Download I. Rates of Return

Chapter 5: Risk and Return: Past and
Prologue
Outline:
I. Rates of Returns
II. Risk and Risk Premiums
III. The Historical Records
Motivation:
- Reality check
I. Rates of Return
Investment: the current commitment of
resources in the expectation of reaping future
benefits that will compensate the investor for (1)
the time the resources are committed, (2) the
expected rate of inflation, and (3) the uncertainty
of the future payoffs.
Essential nature of investment:
1. Reduce current consumption.
2. Planned later consumption.
Typically, economic agents prefer to consume
today than tomorrow. Therefore, incentives
(expected returns) need to be provided to induce
an investment. The rate of exchange between
future consumption (future dollars) and current
consumption (current dollars) is the pure rate of
interest.
Required rate of return
= pure (real) rate of interest  (expected)
inflation rate  risk premium
= nominal risk-free return  risk premium
The symbol  means an aggregation operation,
not an addition operation.
The real risk-free rate, also called pure time
value of money, is the basic interest rate,
assuming no inflation and no uncertainty about
future cash flows.
- Economists tend to believe that there is a
positive relationship between the real
growth rate in the economy and the real riskfree rate.
- The real risk-free rate is widely believed that
it changes only gradually (so it is quite
stable) over the long term.
The risk-free rate that we observe in the market,
e.g., annualized 1-month T-bill rate, is called the
nominal risk-free rate. This nominal rate is an
aggregation of the real rate and expected
inflation rate.
- It is not stable over the long term.
- Two other factors influence the nominal
rate: (1) the relative ease of tightness in the
capital markets, and (2) changes in the
expected rate of inflation.
Example: Suppose that the 12-month T-bill rate
is 9% and the expected rate of inflation is 5%.
What is the real rate?
Nominal risk-free rate = [(1 + real risk-free rate
(1 + expected rate of inflation)]  1
Real risk-free rate = [(1 + nominal risk-free
rate)/(1 + expected rate of inflation)]  1 = [(1 +
9%)/(1+5%)]  1 = 3.8%
Risk premium: the additional return added to the
nominal rate of interest. Risk premium is
required to compensate the investment risk, the
uncertainty of the payoffs from a risky
investment.
- The risk-free asset, e.g., 1-month T-bill, has
a risk premium of zero.
Where does uncertainty come from?
1. Business risk: the uncertainty of income flows
caused by the nature of a firm’s business and
operation.
2. Financial risk: the uncertainty introduced by
the use of debt.
3. Liquidity risk: the possible losses associated
with an immediate liquidation of an asset.
4. Exchange rate risk: the uncertainty in
exchange rate movements.
5. Country (political) risk: the uncertainty of
returns caused by the possibility of a major
change in the political or economic
environment of a country.
6. Timing of payoffs: a potential loss is viewed
to be riskier when the economy is in a poor
state.
Return and Risk can be estimated and measured
using (1) historical return data or (2) subjective
probability distributions.
[1]. Historical returns and risk
Consider an investment with the following
observations:
Year
Beginning Value
1
2
3
100.0
115.0
138.0
Ending Value
(including cash
dividends)
115.0
138.0
110.4
Holding Period: the period during which the
investor owns an investment. In this case, the
holding period is three years.
The 3-year holding period return is:
Holding period return (HPR)
= (ending value/beginning value) − 1 =
(110.4/100.0) − 1
= 0.104 = 10.4%
Frequently, we want to estimate a rate of return
that indicates this investment’s typical
experience and the return rate that we should
expect to receive over the next 12 months. We
often use the mean historical rate of return to
estimate the expected rate of return. To do so,
we first define every year a holding period. We
then calculate HPR for each year:
Year
1
2
3
Beginning Ending
Value
Value
100.0
115.0
115.0
138.0
138.0
110.4
HPR
0.15
0.20
-0.20
Two methods exist for estimating mean
historical return: (1) the arithmetic mean (AM),
and (2) the geometric mean (GM).
AM = (  HPY )/n = (0.15 + 0.20 + -0.20)/3 = 5%
AM is a good indicator of the expected rate of
return for an investment during a future
individual year. However, AM is biased upward
(AM  GM; AM = GM only if rates of return
are the same for all years) if we are attempting
to measure an asset’s long-term performance.
GM = ( 1  HPR )1/n  1
= (1.15  1.20  0.80)1/3  1 = 3.353%.
GM is a better measure for long-term
performance evaluation.
Risk: the uncertainty that an investment will
earn its expected rate of return.
We typically use variance and/or stand deviation
to measure the investment’s (total) risk. If we
estimate variance and standard deviation using
historical return observations:
Variance (i2)
n
=
 ( HPR  AM )
i
2
i 1
n 1
= [(15%  5%)2 + (20%  5%)2 +(-20% 
5%)2]/2 = 0.0476
Standard deviation (i) = (i 2) 1/2 = 21.82%
II. Risk and Risk Premiums
[2]. Risk and return based on a probability
distribution
An investor can form subjective probabilities
based on the historical performance of the
investment or similar investments modified by
the investor’s expectations for the future. For
example, the investor may believe that there are
three possible states (S = 3):
States of
Economics
Good
Fair
Poor
Probability
Rate of Return
15%
70%
15%
20%
10%
-20%
Then the expected rate of return:
E(Ri) =  [ (probabilitys)  (possible returns)]
S
s 1
= [15%  20% + 70%  10% + 15%  -20%]
= 7%
The variance:
Variance (i2)
=  [ (probabilitys)  (possible returns  E(Ri))2]
S
s 1
= 15%  (20%  7%)2 + 70%  (10%  7%)2
+ 15%  (-20%  7%)2
= 0.0141
Standard deviation (i) = (i2) 1/2 =
= 11.87%
0.0141
Risk premium: an expected return in excess of
that on risk-free securities = expected return −
nominal risk-free rate
Excess return: rate of return in excess of the Tbill rate = rate of return − nominal risk-free rate
Risk aversion: reluctance to accept risk
Holding other factors constant, the higher the
risk aversion, the higher the risk premium
III. The Historical Record, 1926-2001
Table 5.3, pp. 140-141
Small stocks (Russell 2000): mean 18.29%, std
39.28%
Large stocks (S&P 500): mean 12.49%, std
20.30%
Long-term bonds: mean 5.53%, std 8.18%
T-bills: mean 3.85%, std 3.25%
Inflation: mean 3.15%, std 4.40%
End-of-chapter problem sets: #1, #2, #3, #4